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Analysis Lab (Hamamuki)
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The lab studies how the faces of a growing crystal move, using partial differential equations.
AboutAbout the lab
Capturing the motion of shapes with equations.
Prof. Hamamuki studies nonlinear partial differential equations.
The main subjects are equations that describe the motion of surfaces (interfaces) separating two different phases of a substance, as they arise in crystal growth and materials science.
Based on the theory of viscosity solutions, a type of weak solution, the lab investigates the existence, uniqueness and long-time behavior of solutions, aiming to give these equations a rigorous mathematical foundation.
Recently, the lab has also become interested in understanding phenomena from multiple perspectives through limit procedures, such as the relationship between the discrete and the continuous and between the micro and the macro.
Key terms
- Partial differential equation
- An equation that relates the rates of change with respect to several variables. It describes how things change.
- Viscosity solution
- One way of defining solutions to differential equations that makes it possible to handle solutions that are not smooth.
- Homogenization theory
- An approach that smooths out fine structures to extract how a system behaves as a whole.
ResearchResearch themes
The lab mainly studies time-evolution equations that describe the motion of interfaces separating two phases, as they appear in crystal growth and materials science. Based on the theory of viscosity solutions, it aims to give these equations a rigorous mathematical foundation.
*Illustrative imageEquations for interface motion
The lab works on time-evolution equations that describe the motion of surfaces (interfaces) separating two different phases of a substance, such as first-order Hamilton–Jacobi equations and second-order curvature flow equations.
*Illustrative imageTheory of viscosity solutions
Based on the theory of viscosity solutions, one concept of weak solutions, the lab studies the introduction of suitable notions of solution, the existence and uniqueness of solutions to initial value problems, and the long-time behavior of solutions.
*Illustrative imageThe discrete and the continuous, the micro and the macro
The lab is also interested in understanding phenomena from multiple perspectives through limit procedures, such as the relationship between the discrete and the continuous and between the micro and the macro (homogenization theory).
MembersFaculty
- Associate ProfessorNao Hamamuki
VoicesStudent voices
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Report an errorAbout the images
The top and research theme images are illustrative and were not taken at the lab. They are cropped from Wikimedia Commons images. Cropped CC BY-SA images are provided under the same CC BY-SA license as the original (version shown on each line).
- Top image: “Real snowflake macro photo” Alexey Kljatov / CC BY-SA 4.0 (cropped)
- Equations for interface motion: “Dendrite in stone from Russia” SKas / CC BY-SA 4.0 (cropped)
- Theory of viscosity solutions: “Plankton creates sea foam 4” Brocken Inaglory / CC BY-SA 4.0 (cropped)
- The discrete and the continuous, the micro and the macro: “Color Metallography of Engineering Alloys” OndrejAmbroz / CC BY 4.0 (cropped)
- Related lab (数理科学系
（眞崎聡） ): “Baltic Sea Wave (Cien Water)” Marek Nowocień / CC BY 2.5 (cropped) - Related lab (解析系
（久保英夫） ): “Surface waves” Roger McLassus / CC BY-SA 3.0 (cropped) - Related lab (数理科学系
（黒田紘敏） ): “Newly fallen snow crystals accumulating on a window sill” W.carter / CC0 (cropped) - Related lab (幾何系
（小林真平） ): “Chen-Gackstatter-Thayer surfaces” Anders Sandberg / CC BY-SA 3.0 (cropped)
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